# (Eran Nevo) The g-Conjecture I

This post is authored by Eran Nevo. (It is the first in a series of five posts.)

Peter McMullen

## The g-conjecture

What are the possible face numbers of triangulations of spheres?

There is only one zero-dimensional sphere and it consists of 2 points.

The one-dimensional triangulations of spheres are simple cycles, having $n$ vertices and $n$ edges, where $n\geq 3$.

The 2-spheres with $n$ vertices have $3n-6$ edges and $2n-4$ triangles, and $n$ can be any integer $\geq 4$. This follows from Euler formula.

For higher-dimensional spheres the number of vertices doesn’t determine all the face numbers, and for spheres of dimension $\geq 5$ a characterization of the possible face numbers is only conjectured! This problem has interesting relations to other mathematical fields, as we shall see. Continue reading